Let be a rational function: where and are holomorphic polynomials. Let be holomorphic on \ and suppose has a pole at each of the points . Finally assume that for all at which these functions are defined. Prove that is a constant multiple of .
You meant, I presume, that are the zeros of the denominator polynomial , right? Well, we get that is an entire function and
as no entire function can dominate another entire function unless one is a scalar multiple of the other (this follows at once from
Liouville's Theorem), we get what we want.